Thermodynamics, Statistical Physics, and Quantum Mechanics

256 SOLUTIONS We take the limit that of the function on the left, and this must equal – since we assumed that the potential vani ...

QUANTUM MECHANICS 257 We equate terms of like powers of The last equation defines The middle equation defines once is known. The ...

258 SOLUTIONS where we used the fact that The probability of energy So the probability of is given by where It is easy to show t ...

QUANTUM MECHANICS 259 d) In order to demonstrate that and are also eigenstates of compose the commutator by (S.5.15.1). Similarl ...

260 SOLUTIONS Up to an arbitrary phase, we see that Starting with the vacuum ket we can write an energy eigenket g) The energysp ...

QUANTUM MECHANICS 261 Since a we have b) We see from (S.5.16.2) that the Hamiltonian is just We demonstrated in (a) that is an e ...

We proceed to find Thus, the result is the same by both approaches. 5.17 Coupled Oscillators (MIT) The Hamiltonian of the system ...

The has a frequency and eigenvalues where is an integer. The oscillator has a frequency of and eigenvalues where is an integer. ...

c) To find the average value of the position operator, we first need to show that Then The expectation value of the position ope ...

c) After the perturbation is added, the Hamiltonian can be solved exactly by completing the square on the where the displacement ...

The only change from the harmonic oscillator for a single spring is that, with two identical springs, the effective spring const ...

b) In order to determine the energy and potential, we operate on the eigen- The constant in the last term can be simplified to I ...

Since we know that Anticipating the result, let Form the raising and lowering operators and Find the commutators Frompart (a) we ...

Expand and apply to (S.5.23.4): Either the state is zero or So Similarly, For and since the only solution is We knew that was re ...

the interaction potential acts only between the electrons, it is natural to write the orbital part in center-of-mass coordinates ...

of any antisymmetric bound state: Any attractive square well has a bound state which is symmetric, but the above condition is re ...

For there are three possible eigenvalues of which gives an energy of For there is one eigenvalue of and this state has zero ener ...

b) From the definition of we deduce that The matrix is the Hermitian conjugate of c) Because and we can construct d) To find the ...

5.28 Constant Matrix Perturbation (Stony Brook) Define where is the eigenvalue. We wish to diagonalize the matrix by finding the ...

To find the amplitude in state we operate on the above equation with The probability P is found from the absolute-magnitude-squa ...